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A319936 Numbers with more than one Collatz tripling step whose odd Collatz trajectory does not contain primes.

%I A319936 #30 May 30 2022 16:33:48
%S A319936 113,227,453,906,909,1812,1813,1818,2417,3624,3626,3636,3637,7248,
%T A319936 7252,7253,7272,7281,9669,14496,14504,14544,14549,14562,14563,19338,
%U A319936 28992,29008,29013,29088,29124,29125,30559,38676,38677,38833,38835,45839,54327,57984
%N A319936 Numbers with more than one Collatz tripling step whose odd Collatz trajectory does not contain primes.
%C A319936 The starting number itself is not counted in the trajectory, otherwise prime numbers like 113 or 227 wouldn't appear in this sequence.
%C A319936 The "odd Collatz trajectory" of a number k is the subset of odd numbers of the full Collatz trajectory of k.
%e A319936 113 is in this sequence because 113*3+1 = 340; 340/2 = 170; 170/2 = 85; 85*3+1 = 256, which goes to 1. The trajectory has 2 (> 1) tripling steps and 85 isn't a prime.
%e A319936 114 is not in this sequence because 114/2 = 57; 57*3+1 = 172; 172/2 = 86; 86/2 = 43, which is a prime, and this trajectory has more than 1 tripling step.
%t A319936 Select[Range[3, 60000], And[Count[#, _?OddQ] > 1, NoneTrue[Rest@ #, PrimeQ]] &@ NestWhileList[If[EvenQ@ #, #/2, 3 # + 1] &, #, # > 2 &, 1, Infinity, -1] &] (* _Michael De Vlieger_, Nov 07 2018 *)
%o A319936 (Java)
%o A319936 for(int i = 0; i < DIM; i++) {
%o A319936     if(!collatzAtLeastOnePrime(c) && collatzTriplingSteps(c) > 1)
%o A319936         System.out.print(c + ", ");
%o A319936 }
%o A319936 boolean collatzAtLeastOnePrime(int i) {
%o A319936     //first step outside the while loop...
%o A319936   if(i % 2 == 0)
%o A319936         i /= 2;
%o A319936     else
%o A319936         i = 3 * i + 1;
%o A319936   //...otherwise prime numbers like 113 or 227 would be excluded
%o A319936     while(i > 1) {
%o A319936         if(i % 2 == 0) {
%o A319936             i /= 2;
%o A319936         }
%o A319936         else {
%o A319936             if(BigInteger.valueOf(i).isProbablePrime(10))
%o A319936                 return true;
%o A319936             i = 3 * i + 1;
%o A319936         }
%o A319936     }
%o A319936     return false;
%o A319936 }
%Y A319936 Cf. A002450, A006577.
%K A319936 nonn
%O A319936 1,1
%A A319936 _Alessandro Polcini_, Oct 10 2018